Journal of Universal Computer Science, vol. 11, no. 12 (2005), 1963-1969 submitted: 14/10/05, accepted: 15/11/05, appeared: 28/12/05 J.UCS On Firmness of the State Space and Positive Elements of a Banach Algebra 1 Robin S. Havea (Department of Mathematics and Computing Science, The University of the South Pacific, PO Box 1168, Suva, Fiji robin.havea@usp.ac.fj) Abstract: This paper is an investigation of positive elements in a Banach algebra. Under the firmness of the state space of a Banach algebra, it is shown that even powers of positive Hermitian elements are in fact positive. Key Words: Banach algebra, constructive mathematics, state space, Hermitian elements, positive elements Category: F.1 1 Introduction The discussion and analysis throughout this paper will be carried out within the framework of Bishop s constructive mathematics 2. It was during the author s postgraduate studies under the guidance and supervision of Bridges that firmness of the space of a Banach algebra was curiously looked at. Bishop and Bridges in the final chapter of [4, page 462] discuss the firmness of the spectrum of Banach algebra. It is a slight translation of that approach that motivated Bridges and the author to use firmness of the state space (which is related to the spectrum) in the investigation of positive elements [9, 13]. It should be pointed out that this short article has two main aims: first, to highlight this interesting aspect of constructive Banach algebra theory, and secondly to stand as one of the testimonies to the many areas where Bridges had been and currently working on. Furthermore, it is the intention of the author that the materials presented in this article would motivate future investigations on constructive Banach algebra theory. The development of constructive Banach algebra theory can be traced back to Bishop s work in [2]. Bishop in the final chapter of [3] shed lights in the constructivisation process and, together with Bridges, topped it with a much smoother development in [4]. For current and recent works on constructive Banach algebra theory,see[5,6,8,10,11,15]. There are two sections that follow immediately after this introductory one. The first contains some technical results and definitions, and the last presents the main results. Additionally, there is a brief discussion of extreme points of a state space and its connection to the character space of the Banach algebra. 1 C. S. Calude, H. Ishihara (eds.). Constructivity, Computability, and Logic. A Collection of Papers in Honour of the 60th Birthday of Douglas Bridges. 2 This is simply mathematics based on intuitionistic logic where existence is strictly interpreted as computability. Details on constructive mathematics can be found in[1,3,4,12].
1964 Havea R.S.: On Firmness of the State Space and Positive Elements... 2 Preliminary We write B to denote a complex unital Banach algebra with identity e and B 1 the unit ball of the dual space B of B. We define the state space of B to be set V B = {f B : f(e) =1= f }. For each t>0theset V t B = {f B : f 1, 1 f(e) t} is a t approximation to V B. A character of B is a bounded homomorphism of B onto C, andthecharacter space (or spectrum) ofb is the set Σ B = {u B : u(e) =1,u(xy) =u(x)u(y) for all x, y B}. Bishop and Bridges [4, page 452] showed that we can t hope in general to prove constructively the compactness of the spectrum. To see this, let (a n ) n=0 be an increasing binary sequence and B the algebra consisting of all sequences x =(x n ) n=0 of complex numbers for which x = (1 a n ) x n (1) n=0 exists. We define the elements x and y =(y n ) n=0 to be equal if x y =0. Then B is a Banach space equipped with norm given by (1). Moreover, if we define the product of any two elements x and y of B by ( n xy =, i=0 x i y n i) n=0 then B is a Banach algebra with identity e =(1, 0, 0,...). Let z = ( 1, 2 1, 2 2, 2 3,... ) B. If a n =1forsomen, then the character space Σ B of B consists of the single element x x 0. On the other hand, if a n =0foralln, then to each complex number ξ with ξ 1 there corresponds an element u ξ of Σ B defined by u ξ (x) = x n ξ n. n=0 Suppose Σ B is compact. Since the mapping u u(z) is uniformly continuous relative to the weak topology on the unit ball of B it maps Σ B to a totally bounded subset of R; whence R =sup{ u(z) : u Σ B } exists. Either R>1orR<2. In the first case, we have a n =0foralln. Inthe second case, we cannot have a n =0foralln. Thus the statement
Havea R.S.: On Firmness of the State Space and Positive Elements... 1965 The spectrum of every separable commutative unital Banach algebra is compact. implies WLPO. Recall that an element f of X, where X of a normed linear space, is normable if its norm f =sup{ f(x) : x 1} exists. If X is separable, and (x n ) is a dense sequence in the unit ball X 1 = {f X : x X ( f (x) x )} of X, then the weak topology on X is induced by the double norm 3, defined by f = 2 n f (x n ) (f X ). Proposition 1. For all but countably many t>0, VB t compactsubsetofb. is a nonempty, weak Proof. Since the mapping f 1 f (e) is uniformly continuous on B 1 relative to the double norm, we see from Theorem 4.9 of [4, page 98] that for all but countably many t>0, V t is either empty or weak compact. An application of Corollary 4.5 of [4, page 341] shows that for such t, V t is nonempty and therefore weak compact. q.e.d. We say that t>0isadmissible if V t B is weak compact. Note that V B = { V t B : t>0 is admissible }, the intersection of a family of nonempty, weak compact sets that is descending in the sense that if 0 <t <t,thenvb t V B t. Being the intersection of a family of complete sets, V B is complete relative to the double norm. We say that V is firm if it is compact and ρ w (V t,v) 0ast 0, where ρ w is the Hausdorff metric on the set of weak compact subsets of B 1. An element x of B is: Hermitian if for each ε>0thereexistst>0 such that Im f(x) <εfor all f V t ; we denote the set of all Hermitian elements of B by Her(B). positive if for each ε>0thereexistst>0 such that Re f(x) ε and Im f(x) <εfor all f V t ;wethenwritex 0. An element f of B is a positive linear functional if f(x) 0foreachpositive element x of B; wethenwritef 0. The following lemma is stated is proved [9]. 3 Double norms defined by different dense sequences in X are equivalent on X 1,and X 1 is weak compact. Moreover, for each x X the mapping f f (x) is uniformly continuous on X 1 with respect to the double norm.
1966 Havea R.S.: On Firmness of the State Space and Positive Elements... Lemma 2. Suppose that the state space of B is firm. Let A be a Banach subalgebra of B, let{x 1,...,x N } be a finitely enumerable subset of A with x 1 = e, and let ε>0. Then there exists an admissible t>0 such that for each f A with f 1 and 1 f (e) t, thereexistsg V A with f (x k ) g (x k ) ε (1 k N) Lemma 3. Let (K λ ) λ L be a nonempty family of totally bounded subsets of a metric space X, andletk = λ L K λ. Suppose that for each ε>0 there exists λ L such that for each x K λ there exists y K with x y <ε.thenk is totally bounded. If also each K λ is complete, then K is compact. Proof. Given ε > 0, choose λ L as in the hypotheses. Let {x 1,...,x N } be a finite ε approximation to K λ,andforeachn choose y n K such that x n y n <ε.lety K K λ. Then there exists n such that y x n <ε and therefore y y n y x n + x n y n <ε+ ε =2ε. Thus {y 1,...,y n } is a 2ε approximation to K. Sinceε > 0 is arbitrary, K is totally bounded. If also each K λ is complete, then K is an intersection of complete sets and so is complete; whence it is compact. q.e.d. 3 Firmness and positivity Proposition 4. If the state space of B is firm, then so is the state space of every separable Banach subalgebra of B. Proof. Let A be a separable Banach subalgebra of B, (x n ) a dense sequence in the unit ball of A, and the corresponding double norm on A.Givenε>0, choose N such that n=n+1 2 n <ε. Using Lemma 2, choose t>0 such that VB t and V A t are weak compact, ρ w (VB t,v B) <ε,and for each f VA t there exists g V A such that f (x k ) g (x k ) ε (1 k N). (2) Let f VA t,andchooseg V A such that (2) holds. We have, in A 1, f g = 2 n (f g)(x n ) = N 2 n f (x n ) g (x n ) + N 2 n ε +2 n=n+1 2 n n=n+1 2 n f (x n ) g (x n ) < 3ε. It follows from Lemma 3 that V A is weak compact. It is then clear from the foregoing that ρ w (VA t,v A) 0ast 0. q.e.d.
Havea R.S.: On Firmness of the State Space and Positive Elements... 1967 Let K be a convex subset of a normed space, and let x 0 K. Wesaythat x 0 is a classical extreme point of K if ( x, y K x 0 = 1 ) 2 (x + y) x = y = x 0 ; an extreme point of K if ( ε >0 δ >0 x, y K x 0 1 ) (x + y) 2 <δ x y <ε. An extreme point is a classical extreme point, and the converse holds classically. The proof of the next result is similar to that given in [14, page 38] for the special case where B is a Banach algebra of functions. This is proved in [9, 13]. Proposition 5. Let A be a commutative, unital Banach algebra generated by Hermitian elements, and K 0 = {f A : f 0, f(e) 1}. Then every classical extreme point of K 0 is an element of Σ B. Lemma 6. For each t (0, 1), if0 <α,β 1 and 1 1 2 (α + β) <t/2, then α>1 t and β>1 t. Proof. If 1 1 2 (α + β) <t/2, then 0 1 2 (1 α)+1 2 (1 β) < t 2, so both 1 2 (1 α) <t/2 and 1 2 (1 β) <t/2. Hence α>1 t and β>1 t. q.e.d. Proposition 7. If the state space V of B is firm, then every extreme point of V is a character of B. Proof. Let be the double norm corresponding to a dense sequence (x n ) in the unit ball of B with x 1 = e. NotingthatV K 0, we show that every extreme point of V is also one of K 0. Accordingly, let f 0 be an extreme point of V,and let ε>0. Choose δ 1 (0,ε) such that if f,g V and 1 2 (f + g) f 0 <δ1, then f g <ε. Then choose an admissible t>0such that ρ w (V t,v) < δ 1 /2. Finally, choose δ 2 > 0 such that if f,g B and f g < δ 2,then f(e) g(e) <t/2. Now let { } 1 δ =min 2 δ 1,δ 2, and consider f,g K 0 with 1 2 (f + g) f 0 <δ.since 1 (f + g)(e) 1 2 = 1 2 (f + g)(e) f 0 (e) < t 2,
1968 Havea R.S.: On Firmness of the State Space and Positive Elements... we have 1 f (e) <tand 1 g(e) <t, by Lemma 6; whence f,g V t,and therefore there exist f,g V such that f f <δ 1 /2and g g <δ 1 /2. We now have 1 2 (f + g ) f 0 1 2 (f + g) f 0 + 1 2 f f + 1 2 g g Hence f g <ε, and therefore < 1 2 δ 1 + 1 4 δ 1 + 1 4 δ 1 = δ 1. f g f f + f g + g g <ε+ δ 1 < 2ε. Since ε>0 is arbitrary, this completes the proof that f 0 is an extreme point, and therefore a classical extreme point, of K 0. By Proposition 5, f 0 is a character of B. q.e.d. Proposition 8. If V is weak compact, then every element of V is a convex combination of characters of B. Proof. It is easily shown that V is convex. An application of the Krein Milman Theorem [4, page 363, (7.5)] shows that V is the closed convex hull of its extreme points; so we can apply Proposition 7. q.e.d. Corollary 9. If the state space of B is firm, then the character space of every separable commutative Banach subalgebra of B is nonempty. Proof. Let A be a separable commutative Banach subalgebra of B. Proposition 4showsthatV A is firm; in particular, it is compact and so has extreme points. By Proposition 7, those extreme points are characters of A. q.e.d. Proposition 10. Let V be firm. Then a Her (B) is positive if and only if f (a) 0 for each f V. Proof. If a is positive, then for each ε>0 there exists an admissible t>0such that Re g(a) ε and Im g(a) <εfor all g V t.iff V,thenf V t and so Re f(a) ε and Im f(a) <ε.sinceε>0 is arbitrary, we conclude that f(a) =Ref (a) 0. Conversely, suppose that f(a) 0foreachf V. Since there exist admissible numbers t>0 such that ρ w (V,V t ) is arbitrarily small, we can choose an admissible t such that for each g V t there exists f V with g(a) f(a) <ε. It now follows that for each g V t, Im g(a) Im f(a) + g(a) f(a) 0+ε = ε and Re g(a) > Re f(a) ε 0 ε = ε. Since ε>0 is arbitrary, we conclude that a 0. q.e.d. Theorem 11. Let a be a Hermitian element of a complex unital Banach algebra B that has firm state space. Then a n is positive for each even positive integer n.
Havea R.S.: On Firmness of the State Space and Positive Elements... 1969 Proof. Let A be a (separable) closed subalgebra of B generated by Hermitian elements a and identity e. By Proposition 4, the state space V A of A is firm. It follows from Proposition 8 that for each f V A there exist characters u 1,...,u m of A, and nonnegative numbers λ 1,...,λ m, such that m i=1 λ i =1 and f m i=1 λ iu i is arbitrary small. In particular, given any n and ε>0, choose u i and λ i such that (f m i=1 λ iu i )(a n ) <ε.ifa 0andn is even, then m Re f (a n ) Re λ i u i (a n m ) f(an ) λ i u i (a n ) i=1 i=1 m Re λ i u i (a) n ε ε, i=1 the last step following from Proposition 10. Since ε>0is arbitrary, we have Re f (a n ) 0foreachf V ; whence, again by Proposition 10, a n 0. q.e.d. References 1. M.J. Beeson, Foundation of Constructive Mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete 6, Springer Verlag, Berlin, 1985. 2. E. Bishop, Constructive methods in the theory of Banach algebras, in Function Algebras (Proceedings of International Symposium on Functional Algebras, Tulane University, Tulane, 1965; F.T. Birtel, ed.), Scott Foresman, Chicago, 343 345, 1966. 3. E. Bishop, Foundations of Constructive Analysis, McGraw Hill, New York, 1967. 4. E. Bishop and D. Bridges, Contructive Analysis, Grundlehren der Mathematischen Wissenschaften 279, Springer Verlag, Berlin, 1985. 5. D. Bridges, Constructive methods in Banach algebra theory, Mathematica Japonica 52(1), 145 161, 2000. 6. D. Bridges, Prime and maximal ideals in constructive ring theory, Comm. Algebra 29(7), 2787 2803, 2001. 7. D. Bridges and R. Havea, A constructive version of the spectral mapping theorem, Math. Logic Q. 47, 299 304, 2001. 8. D. Bridges and R. Havea, Comments on maximal ideals in a Banach algebra, preprint, University of Canterbury, 2005. 9. D. Bridges and R. Havea, Powers of a Hermitian element of a Banach algebra, submitted. 10. D. Bridges, R. Havea, and P. Schuster, Finitely generated Banach algebras and local Nullstellensätze. To appear in Publ. Math. Debrecen. 11. D. Bridges, R. Havea, and P. Schuster, Ideal in constructive Banach algebra theory. Submitted. 12. D. Bridges and F. Richman, Variesties of Constructive Mathematics, London Math. Soc. Lecture Notes 97, Cambridge University Press, Cambridge, 1987. 13. R.S. Havea, Constructive Spectral and Numerical Range Theory, PhDthesis, University of Canterbury, Christchurch, 2001. 14. Richard B. Holmes, Geometric Functional Analysis and its Applications, Graduate Text in Mathematics 24, Springer Verlag, New York, 1975. 15. Ray Mines, Fred Richman and Wim Ruitenburg, A Course in Constructive Algebra, Universitext, Springer Verlag, New York, 1988.